Abstract
Generalizing the realized variance, the realized skewness (Neuberger, 2012) and the realized kurtosis (Bae and Lee, 2020), we construct realized cumulants with the so-called aggregation property. They are unbiased statistics of the cumulants of a martingale marginal based on sub-period increments of the martingale and its lower-order conditional cumulant processes. Our key finding is a relation between the aggregation property and the complete Bell polynomials. For an application we give an alternative proof and an extension of a cumulant recursion formula recently obtained by Lacoin et al. (2019) and Friz et al. (2020).
Highlights
A square-integrable martingale M = {Mt}t∈[0,T ] satisfies NE[(MT − M0)n] = E (Mtj − Mtj−1 )n j=1 (1.1)for n = 2 with an arbitrary time partition 0 = t0 < · · · < tN = T
Generalizing the realized variance, the realized skewness (Neuberger, 2012) and the realized kurtosis (Bae and Lee, 2020), we construct realized cumulants with the so-called aggregation property. They are unbiased statistics of the cumulants of a martingale marginal based on sub-period increments of the martingale and its lower-order conditional cumulant processes
For an application we give an alternative proof and an extension of a cumulant recursion formula recently obtained by Lacoin et al (2019) and Friz et al (2020)
Summary
Bae and Lee [3] further extended the idea to construct the realized kurtosis, by finding that the aggregation property is met by g(x, y, z) = x4 + 6x2y + 3y2 + 4xz and X = (M, M (2), M (3)). Neuberger [10] and Bae and Lee [3]’s approach to find those polynomials is rather brute force. They showed that there is no other analytic function (up to linear combinations) g satisfying the aggregation property with X = (M, M (2), M (3)).
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